Four objects, each with a positive charge of 1.0 3 1025 C,
are placed at the corners of a 45° rhombus with sides of
length 1.0 m, as in Figure 13. Calculate the magnitude of
the net force on each charge
Figure 13.
an equilateral parallelogram
To calculate the magnitude of the net force on each charge in a rhombus, we can use Coulomb's Law. According to Coulomb's Law, the force between two charges is given by the equation:
F = (k * q1 * q2) / r²
where F is the force between the charges, k is the electrostatic constant (9.0 × 10^9 N•m²/C²), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
In this case, we have four charges, each with a positive charge of 1.0 × 10^25 C. Since the charges are all the same, we can calculate the force between any two adjacent charges in the rhombus and then multiply it by the appropriate cosine or sine value to find the net force on each charge.
Let's consider the forces acting on one of the charges. If we draw a diagram of the rhombus, we can see that each charge has two adjacent charges and two opposite charges.
The adjacent charges are located on the sides of the rhombus and are a distance of 1.0 m away from the central charge. The opposite charges are located at the corners of the rhombus and are diagonally across from the central charge. The distance between the opposite charges can be found using the Pythagorean theorem.
To calculate the net force on each charge, we need to find the forces from the adjacent charges and then resolve them into horizontal and vertical components. The forces from the opposite charges are already in the correct direction.
Let's denote the net force on each charge as F_net. To find F_net, we can follow these steps:
1. Calculate the force between each adjacent charge and the central charge using Coulomb's Law:
F_adjacent = (k * q * q) / r²
where q is the charge (1.0 × 10^25 C) and r is the distance between the adjacent charges (1.0 m).
2. Calculate the force between each opposite charge and the central charge using Coulomb's Law. The distance between the opposite charges can be found using the Pythagorean theorem:
r_opposite = √(r² + r²)
where r is the distance between adjacent charges (1.0 m).
F_opposite = (k * q * q) / r_opposite²
3. Find the horizontal and vertical components of the forces from the adjacent charges. Since the angle between the adjacent charges and the horizontal is 45°, we can use the trigonometric formula:
F_horizontal = F_adjacent * cos(45°)
F_vertical = F_adjacent * sin(45°)
4. Calculate the net force on each charge by adding up the horizontal and vertical components of the forces from both the adjacent and opposite charges:
F_net = (F_horizontal_adjacent + F_horizontal_opposite)² + (F_vertical_adjacent + F_vertical_opposite)²
Now you can substitute the values into these equations and calculate the magnitude of the net force on each charge.